Spatial effects in low neutron source start-up and associated stochastic phenomena

Highlights: •Extension of Pál-Bell equation to calculate the effect of source position on source multiplier. •Use of Chebyshev-Gauss-Lobatto collocation methods. •Examples of Hurwitz curves for sources in a sphere. •Use of Gamma pdf to illustrate the behaviour of the neutron population. -- Abstract: This work concerns the calculation of the neutron source strength necessary to start up a nuclear reactor such that the likelihood of an undesirable stochastic transient is reduced to a specified value (e.g. 10⁻⁸). We extend our earlier point model work on low source calculations to include the spatial variation of the neutron source. Results for the source multiplier for a given safety factor are obtained for slab, cylindrical and spherical systems. The spatial term in the Pál-Bell equation is dealt with by Chebyshev-Gauss-Lobatto collocation methods and this enables an extrapolation distance to be included, thereby simulating a reflector. Results are given for a range of system sizes, and corresponding source multipliers for safe source determination are obtained. The saddlepoint method is used to invert the generating function. In addition to the low source calculations, we have also tested the collocation method on the survival probability in a sphere which demonstrates excellent convergence. We also comment on the usefulness of the Gamma pdf for spatially dependent problems. For clarity of presentation, some of the detailed mathematical work is relegated to Appendices.